# On solutions of the transport equation in the presence of singularities

**Authors:** Evelyne Miot, Nicholas Sharples  
**arXiv:** [2006.09200](https://arxiv.org/abs/2006.09200)  
**Date:** 2020-06-16  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2006.09200v1/

## Key points

- The paper addresses the transport equation in the presence of singularities
- Solutions to this transport equation exist and are unique under certain conditions
- The authors relax the condition of bounded variation everywhere for the vector field
- Almost every trajectory of the associated regular Lagrangian flow does not intersect the set of singularities
- The paper considers a specific case where an initial set of singularities evolves in time, resulting in singularities consisting of curves in phase space
- Solutions to the transport equation still exist and are unique in this case, under certain conditions
- The results have potential applications in fluid dynamics and mathematical physics.

## Summary

The paper titled "On solutions of the transport equation in the presence of singularities" addresses the transport equation on $[0,T]\times \mathbb{R}^n$ in the presence of singularities. The authors consider a situation where the vector field is bounded variation (BV) off a set $S\subset [0,T]\times \mathbb{R}^n$. They demonstrate that solutions to this transport equation exist and are unique, provided that the set of singularities has a sufficiently small anisotropic fractal dimension and the normal component of the vector field is sufficiently integrable near these singularities. This result is an improvement upon previous findings by Ambrosio, who required the vector field to be of bounded variation everywhere. By relaxing this condition, the authors provide a more general framework for studying solutions to the transport equation. Furthermore, the authors show that under these conditions, almost every trajectory of the associated regular Lagrangian flow does not intersect the set $S$ of singularities. This observation has important implications for understanding and analyzing fluid dynamics phenomena. In addition to these general results, the paper also considers a specific case where an initial set of singularities evolves in time, resulting in singularities consisting of curves in phase space. This scenario is common in applications such as vortex dynamics. The authors demonstrate that solutions to the transport equation still exist and are unique in this case, provided that the box-counting dimension of these evolving singularities is bounded in terms of their H\"older exponent. Overall, this paper provides valuable insights into solutions of the transport equation when dealing with singularities. The results presented have potential applications in various fields including fluid dynamics and mathematical physics.